Properties

Label 1600.2.o.k.607.2
Level $1600$
Weight $2$
Character 1600.607
Analytic conductor $12.776$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(543,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.543");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.o (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.12745506816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 23x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 607.2
Root \(1.54779 + 1.54779i\) of defining polynomial
Character \(\chi\) \(=\) 1600.607
Dual form 1600.2.o.k.543.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.87083 + 1.87083i) q^{3} +(2.44949 - 2.44949i) q^{7} -4.00000i q^{9} +O(q^{10})\) \(q+(-1.87083 + 1.87083i) q^{3} +(2.44949 - 2.44949i) q^{7} -4.00000i q^{9} +4.58258 q^{11} +(-3.74166 - 3.74166i) q^{13} +(-1.22474 - 1.22474i) q^{17} +4.58258i q^{19} +9.16515i q^{21} +(-4.89898 - 4.89898i) q^{23} +(1.87083 + 1.87083i) q^{27} -9.16515 q^{29} -4.00000i q^{31} +(-8.57321 + 8.57321i) q^{33} +(-3.74166 + 3.74166i) q^{37} +14.0000 q^{39} -9.00000 q^{41} +(-3.74166 + 3.74166i) q^{43} +(-2.44949 + 2.44949i) q^{47} -5.00000i q^{49} +4.58258 q^{51} +(-8.57321 - 8.57321i) q^{57} -9.16515i q^{61} +(-9.79796 - 9.79796i) q^{63} +(-1.87083 - 1.87083i) q^{67} +18.3303 q^{69} +6.00000i q^{71} +(-6.12372 + 6.12372i) q^{73} +(11.2250 - 11.2250i) q^{77} +10.0000 q^{79} +5.00000 q^{81} +(5.61249 - 5.61249i) q^{83} +(17.1464 - 17.1464i) q^{87} -15.0000i q^{89} -18.3303 q^{91} +(7.48331 + 7.48331i) q^{93} +(-4.89898 - 4.89898i) q^{97} -18.3303i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 112 q^{39} - 72 q^{41} + 80 q^{79} + 40 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.87083 + 1.87083i −1.08012 + 1.08012i −0.0836263 + 0.996497i \(0.526650\pi\)
−0.996497 + 0.0836263i \(0.973350\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.44949 2.44949i 0.925820 0.925820i −0.0716124 0.997433i \(-0.522814\pi\)
0.997433 + 0.0716124i \(0.0228145\pi\)
\(8\) 0 0
\(9\) 4.00000i 1.33333i
\(10\) 0 0
\(11\) 4.58258 1.38170 0.690849 0.722999i \(-0.257237\pi\)
0.690849 + 0.722999i \(0.257237\pi\)
\(12\) 0 0
\(13\) −3.74166 3.74166i −1.03775 1.03775i −0.999259 0.0384901i \(-0.987745\pi\)
−0.0384901 0.999259i \(-0.512255\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.22474 1.22474i −0.297044 0.297044i 0.542811 0.839855i \(-0.317360\pi\)
−0.839855 + 0.542811i \(0.817360\pi\)
\(18\) 0 0
\(19\) 4.58258i 1.05131i 0.850696 + 0.525657i \(0.176181\pi\)
−0.850696 + 0.525657i \(0.823819\pi\)
\(20\) 0 0
\(21\) 9.16515i 2.00000i
\(22\) 0 0
\(23\) −4.89898 4.89898i −1.02151 1.02151i −0.999764 0.0217443i \(-0.993078\pi\)
−0.0217443 0.999764i \(-0.506922\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.87083 + 1.87083i 0.360041 + 0.360041i
\(28\) 0 0
\(29\) −9.16515 −1.70193 −0.850963 0.525226i \(-0.823981\pi\)
−0.850963 + 0.525226i \(0.823981\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i −0.933257 0.359211i \(-0.883046\pi\)
0.933257 0.359211i \(-0.116954\pi\)
\(32\) 0 0
\(33\) −8.57321 + 8.57321i −1.49241 + 1.49241i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.74166 + 3.74166i −0.615125 + 0.615125i −0.944277 0.329152i \(-0.893237\pi\)
0.329152 + 0.944277i \(0.393237\pi\)
\(38\) 0 0
\(39\) 14.0000 2.24179
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −3.74166 + 3.74166i −0.570597 + 0.570597i −0.932295 0.361698i \(-0.882197\pi\)
0.361698 + 0.932295i \(0.382197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.44949 + 2.44949i −0.357295 + 0.357295i −0.862815 0.505520i \(-0.831301\pi\)
0.505520 + 0.862815i \(0.331301\pi\)
\(48\) 0 0
\(49\) 5.00000i 0.714286i
\(50\) 0 0
\(51\) 4.58258 0.641689
\(52\) 0 0
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −8.57321 8.57321i −1.13555 1.13555i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 9.16515i 1.17348i −0.809776 0.586739i \(-0.800412\pi\)
0.809776 0.586739i \(-0.199588\pi\)
\(62\) 0 0
\(63\) −9.79796 9.79796i −1.23443 1.23443i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.87083 1.87083i −0.228558 0.228558i 0.583532 0.812090i \(-0.301670\pi\)
−0.812090 + 0.583532i \(0.801670\pi\)
\(68\) 0 0
\(69\) 18.3303 2.20671
\(70\) 0 0
\(71\) 6.00000i 0.712069i 0.934473 + 0.356034i \(0.115871\pi\)
−0.934473 + 0.356034i \(0.884129\pi\)
\(72\) 0 0
\(73\) −6.12372 + 6.12372i −0.716728 + 0.716728i −0.967934 0.251206i \(-0.919173\pi\)
0.251206 + 0.967934i \(0.419173\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.2250 11.2250i 1.27920 1.27920i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 5.00000 0.555556
\(82\) 0 0
\(83\) 5.61249 5.61249i 0.616050 0.616050i −0.328466 0.944516i \(-0.606531\pi\)
0.944516 + 0.328466i \(0.106531\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 17.1464 17.1464i 1.83829 1.83829i
\(88\) 0 0
\(89\) 15.0000i 1.59000i −0.606612 0.794998i \(-0.707472\pi\)
0.606612 0.794998i \(-0.292528\pi\)
\(90\) 0 0
\(91\) −18.3303 −1.92154
\(92\) 0 0
\(93\) 7.48331 + 7.48331i 0.775984 + 0.775984i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.89898 4.89898i −0.497416 0.497416i 0.413217 0.910633i \(-0.364405\pi\)
−0.910633 + 0.413217i \(0.864405\pi\)
\(98\) 0 0
\(99\) 18.3303i 1.84226i
\(100\) 0 0
\(101\) 9.16515i 0.911967i −0.889988 0.455983i \(-0.849288\pi\)
0.889988 0.455983i \(-0.150712\pi\)
\(102\) 0 0
\(103\) 2.44949 + 2.44949i 0.241355 + 0.241355i 0.817411 0.576055i \(-0.195409\pi\)
−0.576055 + 0.817411i \(0.695409\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.61249 5.61249i −0.542580 0.542580i 0.381705 0.924284i \(-0.375337\pi\)
−0.924284 + 0.381705i \(0.875337\pi\)
\(108\) 0 0
\(109\) −9.16515 −0.877862 −0.438931 0.898521i \(-0.644643\pi\)
−0.438931 + 0.898521i \(0.644643\pi\)
\(110\) 0 0
\(111\) 14.0000i 1.32882i
\(112\) 0 0
\(113\) −1.22474 + 1.22474i −0.115214 + 0.115214i −0.762363 0.647149i \(-0.775961\pi\)
0.647149 + 0.762363i \(0.275961\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −14.9666 + 14.9666i −1.38367 + 1.38367i
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 10.0000 0.909091
\(122\) 0 0
\(123\) 16.8375 16.8375i 1.51818 1.51818i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(128\) 0 0
\(129\) 14.0000i 1.23263i
\(130\) 0 0
\(131\) −18.3303 −1.60153 −0.800763 0.598981i \(-0.795572\pi\)
−0.800763 + 0.598981i \(0.795572\pi\)
\(132\) 0 0
\(133\) 11.2250 + 11.2250i 0.973329 + 0.973329i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.0227 + 11.0227i 0.941733 + 0.941733i 0.998394 0.0566604i \(-0.0180452\pi\)
−0.0566604 + 0.998394i \(0.518045\pi\)
\(138\) 0 0
\(139\) 4.58258i 0.388689i 0.980933 + 0.194344i \(0.0622580\pi\)
−0.980933 + 0.194344i \(0.937742\pi\)
\(140\) 0 0
\(141\) 9.16515i 0.771845i
\(142\) 0 0
\(143\) −17.1464 17.1464i −1.43386 1.43386i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 9.35414 + 9.35414i 0.771517 + 0.771517i
\(148\) 0 0
\(149\) 9.16515 0.750838 0.375419 0.926855i \(-0.377499\pi\)
0.375419 + 0.926855i \(0.377499\pi\)
\(150\) 0 0
\(151\) 10.0000i 0.813788i 0.913475 + 0.406894i \(0.133388\pi\)
−0.913475 + 0.406894i \(0.866612\pi\)
\(152\) 0 0
\(153\) −4.89898 + 4.89898i −0.396059 + 0.396059i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.9666 14.9666i 1.19447 1.19447i 0.218668 0.975799i \(-0.429829\pi\)
0.975799 0.218668i \(-0.0701711\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) 1.87083 1.87083i 0.146535 0.146535i −0.630033 0.776568i \(-0.716959\pi\)
0.776568 + 0.630033i \(0.216959\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.34847 + 7.34847i −0.568642 + 0.568642i −0.931748 0.363106i \(-0.881716\pi\)
0.363106 + 0.931748i \(0.381716\pi\)
\(168\) 0 0
\(169\) 15.0000i 1.15385i
\(170\) 0 0
\(171\) 18.3303 1.40175
\(172\) 0 0
\(173\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.7477i 1.02755i 0.857924 + 0.513777i \(0.171754\pi\)
−0.857924 + 0.513777i \(0.828246\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 17.1464 + 17.1464i 1.26750 + 1.26750i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −5.61249 5.61249i −0.410426 0.410426i
\(188\) 0 0
\(189\) 9.16515 0.666667
\(190\) 0 0
\(191\) 24.0000i 1.73658i −0.496058 0.868290i \(-0.665220\pi\)
0.496058 0.868290i \(-0.334780\pi\)
\(192\) 0 0
\(193\) 6.12372 6.12372i 0.440795 0.440795i −0.451484 0.892279i \(-0.649105\pi\)
0.892279 + 0.451484i \(0.149105\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 7.00000 0.493742
\(202\) 0 0
\(203\) −22.4499 + 22.4499i −1.57568 + 1.57568i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −19.5959 + 19.5959i −1.36201 + 1.36201i
\(208\) 0 0
\(209\) 21.0000i 1.45260i
\(210\) 0 0
\(211\) 13.7477 0.946433 0.473216 0.880946i \(-0.343093\pi\)
0.473216 + 0.880946i \(0.343093\pi\)
\(212\) 0 0
\(213\) −11.2250 11.2250i −0.769122 0.769122i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −9.79796 9.79796i −0.665129 0.665129i
\(218\) 0 0
\(219\) 22.9129i 1.54831i
\(220\) 0 0
\(221\) 9.16515i 0.616515i
\(222\) 0 0
\(223\) 4.89898 + 4.89898i 0.328060 + 0.328060i 0.851848 0.523788i \(-0.175482\pi\)
−0.523788 + 0.851848i \(0.675482\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.2250 11.2250i −0.745028 0.745028i 0.228513 0.973541i \(-0.426614\pi\)
−0.973541 + 0.228513i \(0.926614\pi\)
\(228\) 0 0
\(229\) −18.3303 −1.21130 −0.605650 0.795731i \(-0.707087\pi\)
−0.605650 + 0.795731i \(0.707087\pi\)
\(230\) 0 0
\(231\) 42.0000i 2.76340i
\(232\) 0 0
\(233\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −18.7083 + 18.7083i −1.21523 + 1.21523i
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 7.00000 0.450910 0.225455 0.974254i \(-0.427613\pi\)
0.225455 + 0.974254i \(0.427613\pi\)
\(242\) 0 0
\(243\) −14.9666 + 14.9666i −0.960110 + 0.960110i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 17.1464 17.1464i 1.09100 1.09100i
\(248\) 0 0
\(249\) 21.0000i 1.33082i
\(250\) 0 0
\(251\) −4.58258 −0.289250 −0.144625 0.989487i \(-0.546198\pi\)
−0.144625 + 0.989487i \(0.546198\pi\)
\(252\) 0 0
\(253\) −22.4499 22.4499i −1.41142 1.41142i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.89898 + 4.89898i 0.305590 + 0.305590i 0.843196 0.537606i \(-0.180671\pi\)
−0.537606 + 0.843196i \(0.680671\pi\)
\(258\) 0 0
\(259\) 18.3303i 1.13899i
\(260\) 0 0
\(261\) 36.6606i 2.26923i
\(262\) 0 0
\(263\) 14.6969 + 14.6969i 0.906252 + 0.906252i 0.995967 0.0897154i \(-0.0285957\pi\)
−0.0897154 + 0.995967i \(0.528596\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 28.0624 + 28.0624i 1.71739 + 1.71739i
\(268\) 0 0
\(269\) −18.3303 −1.11762 −0.558809 0.829296i \(-0.688742\pi\)
−0.558809 + 0.829296i \(0.688742\pi\)
\(270\) 0 0
\(271\) 8.00000i 0.485965i 0.970031 + 0.242983i \(0.0781258\pi\)
−0.970031 + 0.242983i \(0.921874\pi\)
\(272\) 0 0
\(273\) 34.2929 34.2929i 2.07550 2.07550i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 18.7083 18.7083i 1.12407 1.12407i 0.132949 0.991123i \(-0.457555\pi\)
0.991123 0.132949i \(-0.0424447\pi\)
\(278\) 0 0
\(279\) −16.0000 −0.957895
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 9.35414 9.35414i 0.556046 0.556046i −0.372133 0.928179i \(-0.621374\pi\)
0.928179 + 0.372133i \(0.121374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −22.0454 + 22.0454i −1.30130 + 1.30130i
\(288\) 0 0
\(289\) 14.0000i 0.823529i
\(290\) 0 0
\(291\) 18.3303 1.07454
\(292\) 0 0
\(293\) −11.2250 11.2250i −0.655770 0.655770i 0.298606 0.954376i \(-0.403478\pi\)
−0.954376 + 0.298606i \(0.903478\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 8.57321 + 8.57321i 0.497468 + 0.497468i
\(298\) 0 0
\(299\) 36.6606i 2.12014i
\(300\) 0 0
\(301\) 18.3303i 1.05654i
\(302\) 0 0
\(303\) 17.1464 + 17.1464i 0.985037 + 0.985037i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −9.35414 9.35414i −0.533869 0.533869i 0.387852 0.921722i \(-0.373217\pi\)
−0.921722 + 0.387852i \(0.873217\pi\)
\(308\) 0 0
\(309\) −9.16515 −0.521387
\(310\) 0 0
\(311\) 18.0000i 1.02069i −0.859971 0.510343i \(-0.829518\pi\)
0.859971 0.510343i \(-0.170482\pi\)
\(312\) 0 0
\(313\) 14.6969 14.6969i 0.830720 0.830720i −0.156895 0.987615i \(-0.550148\pi\)
0.987615 + 0.156895i \(0.0501485\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.4499 + 22.4499i −1.26091 + 1.26091i −0.310264 + 0.950650i \(0.600417\pi\)
−0.950650 + 0.310264i \(0.899583\pi\)
\(318\) 0 0
\(319\) −42.0000 −2.35155
\(320\) 0 0
\(321\) 21.0000 1.17211
\(322\) 0 0
\(323\) 5.61249 5.61249i 0.312287 0.312287i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 17.1464 17.1464i 0.948200 0.948200i
\(328\) 0 0
\(329\) 12.0000i 0.661581i
\(330\) 0 0
\(331\) 32.0780 1.76317 0.881584 0.472027i \(-0.156478\pi\)
0.881584 + 0.472027i \(0.156478\pi\)
\(332\) 0 0
\(333\) 14.9666 + 14.9666i 0.820166 + 0.820166i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.3712 + 18.3712i 1.00074 + 1.00074i 1.00000 0.000741840i \(0.000236135\pi\)
0.000741840 1.00000i \(0.499764\pi\)
\(338\) 0 0
\(339\) 4.58258i 0.248891i
\(340\) 0 0
\(341\) 18.3303i 0.992642i
\(342\) 0 0
\(343\) 4.89898 + 4.89898i 0.264520 + 0.264520i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5.61249 5.61249i −0.301294 0.301294i 0.540226 0.841520i \(-0.318339\pi\)
−0.841520 + 0.540226i \(0.818339\pi\)
\(348\) 0 0
\(349\) 9.16515 0.490599 0.245300 0.969447i \(-0.421114\pi\)
0.245300 + 0.969447i \(0.421114\pi\)
\(350\) 0 0
\(351\) 14.0000i 0.747265i
\(352\) 0 0
\(353\) −24.4949 + 24.4949i −1.30373 + 1.30373i −0.377875 + 0.925856i \(0.623345\pi\)
−0.925856 + 0.377875i \(0.876655\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 11.2250 11.2250i 0.594089 0.594089i
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) −2.00000 −0.105263
\(362\) 0 0
\(363\) −18.7083 + 18.7083i −0.981930 + 0.981930i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −14.6969 + 14.6969i −0.767174 + 0.767174i −0.977608 0.210434i \(-0.932512\pi\)
0.210434 + 0.977608i \(0.432512\pi\)
\(368\) 0 0
\(369\) 36.0000i 1.87409i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 14.9666 + 14.9666i 0.774943 + 0.774943i 0.978966 0.204023i \(-0.0654018\pi\)
−0.204023 + 0.978966i \(0.565402\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 34.2929 + 34.2929i 1.76617 + 1.76617i
\(378\) 0 0
\(379\) 13.7477i 0.706173i −0.935591 0.353087i \(-0.885132\pi\)
0.935591 0.353087i \(-0.114868\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2.44949 2.44949i −0.125163 0.125163i 0.641750 0.766914i \(-0.278208\pi\)
−0.766914 + 0.641750i \(0.778208\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 14.9666 + 14.9666i 0.760797 + 0.760797i
\(388\) 0 0
\(389\) −18.3303 −0.929383 −0.464692 0.885473i \(-0.653835\pi\)
−0.464692 + 0.885473i \(0.653835\pi\)
\(390\) 0 0
\(391\) 12.0000i 0.606866i
\(392\) 0 0
\(393\) 34.2929 34.2929i 1.72985 1.72985i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −7.48331 + 7.48331i −0.375577 + 0.375577i −0.869504 0.493927i \(-0.835561\pi\)
0.493927 + 0.869504i \(0.335561\pi\)
\(398\) 0 0
\(399\) −42.0000 −2.10263
\(400\) 0 0
\(401\) −15.0000 −0.749064 −0.374532 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) −14.9666 + 14.9666i −0.745541 + 0.745541i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −17.1464 + 17.1464i −0.849917 + 0.849917i
\(408\) 0 0
\(409\) 17.0000i 0.840596i −0.907386 0.420298i \(-0.861926\pi\)
0.907386 0.420298i \(-0.138074\pi\)
\(410\) 0 0
\(411\) −41.2432 −2.03438
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −8.57321 8.57321i −0.419832 0.419832i
\(418\) 0 0
\(419\) 22.9129i 1.11937i −0.828706 0.559684i \(-0.810923\pi\)
0.828706 0.559684i \(-0.189077\pi\)
\(420\) 0 0
\(421\) 36.6606i 1.78673i −0.449333 0.893364i \(-0.648338\pi\)
0.449333 0.893364i \(-0.351662\pi\)
\(422\) 0 0
\(423\) 9.79796 + 9.79796i 0.476393 + 0.476393i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −22.4499 22.4499i −1.08643 1.08643i
\(428\) 0 0
\(429\) 64.1561 3.09748
\(430\) 0 0
\(431\) 30.0000i 1.44505i −0.691345 0.722525i \(-0.742982\pi\)
0.691345 0.722525i \(-0.257018\pi\)
\(432\) 0 0
\(433\) 13.4722 13.4722i 0.647432 0.647432i −0.304939 0.952372i \(-0.598636\pi\)
0.952372 + 0.304939i \(0.0986362\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 22.4499 22.4499i 1.07393 1.07393i
\(438\) 0 0
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) −20.0000 −0.952381
\(442\) 0 0
\(443\) 28.0624 28.0624i 1.33329 1.33329i 0.430874 0.902412i \(-0.358205\pi\)
0.902412 0.430874i \(-0.141795\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −17.1464 + 17.1464i −0.810998 + 0.810998i
\(448\) 0 0
\(449\) 27.0000i 1.27421i 0.770778 + 0.637104i \(0.219868\pi\)
−0.770778 + 0.637104i \(0.780132\pi\)
\(450\) 0 0
\(451\) −41.2432 −1.94207
\(452\) 0 0
\(453\) −18.7083 18.7083i −0.878992 0.878992i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −25.7196 25.7196i −1.20311 1.20311i −0.973214 0.229900i \(-0.926160\pi\)
−0.229900 0.973214i \(-0.573840\pi\)
\(458\) 0 0
\(459\) 4.58258i 0.213896i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 9.79796 + 9.79796i 0.455350 + 0.455350i 0.897126 0.441776i \(-0.145651\pi\)
−0.441776 + 0.897126i \(0.645651\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.2250 + 11.2250i 0.519430 + 0.519430i 0.917399 0.397969i \(-0.130285\pi\)
−0.397969 + 0.917399i \(0.630285\pi\)
\(468\) 0 0
\(469\) −9.16515 −0.423207
\(470\) 0 0
\(471\) 56.0000i 2.58034i
\(472\) 0 0
\(473\) −17.1464 + 17.1464i −0.788394 + 0.788394i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 28.0000 1.27669
\(482\) 0 0
\(483\) 44.8999 44.8999i 2.04302 2.04302i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 14.6969 14.6969i 0.665982 0.665982i −0.290802 0.956783i \(-0.593922\pi\)
0.956783 + 0.290802i \(0.0939219\pi\)
\(488\) 0 0
\(489\) 7.00000i 0.316551i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 11.2250 + 11.2250i 0.505547 + 0.505547i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14.6969 + 14.6969i 0.659248 + 0.659248i
\(498\) 0 0
\(499\) 18.3303i 0.820577i −0.911956 0.410289i \(-0.865428\pi\)
0.911956 0.410289i \(-0.134572\pi\)
\(500\) 0 0
\(501\) 27.4955i 1.22841i
\(502\) 0 0
\(503\) 9.79796 + 9.79796i 0.436869 + 0.436869i 0.890957 0.454088i \(-0.150035\pi\)
−0.454088 + 0.890957i \(0.650035\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −28.0624 28.0624i −1.24630 1.24630i
\(508\) 0 0
\(509\) 18.3303 0.812476 0.406238 0.913767i \(-0.366840\pi\)
0.406238 + 0.913767i \(0.366840\pi\)
\(510\) 0 0
\(511\) 30.0000i 1.32712i
\(512\) 0 0
\(513\) −8.57321 + 8.57321i −0.378517 + 0.378517i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −11.2250 + 11.2250i −0.493674 + 0.493674i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) 9.35414 9.35414i 0.409028 0.409028i −0.472371 0.881400i \(-0.656602\pi\)
0.881400 + 0.472371i \(0.156602\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.89898 + 4.89898i −0.213403 + 0.213403i
\(528\) 0 0
\(529\) 25.0000i 1.08696i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 33.6749 + 33.6749i 1.45862 + 1.45862i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −25.7196 25.7196i −1.10988 1.10988i
\(538\) 0 0
\(539\) 22.9129i 0.986928i
\(540\) 0 0
\(541\) 18.3303i 0.788081i 0.919093 + 0.394041i \(0.128923\pi\)
−0.919093 + 0.394041i \(0.871077\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 13.0958 + 13.0958i 0.559936 + 0.559936i 0.929289 0.369353i \(-0.120421\pi\)
−0.369353 + 0.929289i \(0.620421\pi\)
\(548\) 0 0
\(549\) −36.6606 −1.56464
\(550\) 0 0
\(551\) 42.0000i 1.78926i
\(552\) 0 0
\(553\) 24.4949 24.4949i 1.04163 1.04163i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11.2250 11.2250i 0.475617 0.475617i −0.428110 0.903727i \(-0.640820\pi\)
0.903727 + 0.428110i \(0.140820\pi\)
\(558\) 0 0
\(559\) 28.0000 1.18427
\(560\) 0 0
\(561\) 21.0000 0.886621
\(562\) 0 0
\(563\) −11.2250 + 11.2250i −0.473076 + 0.473076i −0.902909 0.429832i \(-0.858573\pi\)
0.429832 + 0.902909i \(0.358573\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 12.2474 12.2474i 0.514344 0.514344i
\(568\) 0 0
\(569\) 3.00000i 0.125767i −0.998021 0.0628833i \(-0.979970\pi\)
0.998021 0.0628833i \(-0.0200296\pi\)
\(570\) 0 0
\(571\) −36.6606 −1.53420 −0.767099 0.641528i \(-0.778301\pi\)
−0.767099 + 0.641528i \(0.778301\pi\)
\(572\) 0 0
\(573\) 44.8999 + 44.8999i 1.87572 + 1.87572i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −28.1691 28.1691i −1.17270 1.17270i −0.981564 0.191132i \(-0.938784\pi\)
−0.191132 0.981564i \(-0.561216\pi\)
\(578\) 0 0
\(579\) 22.9129i 0.952227i
\(580\) 0 0
\(581\) 27.4955i 1.14070i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.0624 + 28.0624i 1.15826 + 1.15826i 0.984850 + 0.173411i \(0.0554789\pi\)
0.173411 + 0.984850i \(0.444521\pi\)
\(588\) 0 0
\(589\) 18.3303 0.755287
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.12372 + 6.12372i −0.251471 + 0.251471i −0.821574 0.570102i \(-0.806904\pi\)
0.570102 + 0.821574i \(0.306904\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −7.48331 + 7.48331i −0.306272 + 0.306272i
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −35.0000 −1.42768 −0.713840 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) 0 0
\(603\) −7.48331 + 7.48331i −0.304744 + 0.304744i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 19.5959 19.5959i 0.795374 0.795374i −0.186988 0.982362i \(-0.559873\pi\)
0.982362 + 0.186988i \(0.0598727\pi\)
\(608\) 0 0
\(609\) 84.0000i 3.40385i
\(610\) 0 0
\(611\) 18.3303 0.741565
\(612\) 0 0
\(613\) −7.48331 7.48331i −0.302248 0.302248i 0.539645 0.841893i \(-0.318559\pi\)
−0.841893 + 0.539645i \(0.818559\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 24.4949 + 24.4949i 0.986127 + 0.986127i 0.999905 0.0137776i \(-0.00438570\pi\)
−0.0137776 + 0.999905i \(0.504386\pi\)
\(618\) 0 0
\(619\) 36.6606i 1.47351i 0.676158 + 0.736757i \(0.263644\pi\)
−0.676158 + 0.736757i \(0.736356\pi\)
\(620\) 0 0
\(621\) 18.3303i 0.735570i
\(622\) 0 0
\(623\) −36.7423 36.7423i −1.47205 1.47205i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −39.2874 39.2874i −1.56899 1.56899i
\(628\) 0 0
\(629\) 9.16515 0.365439
\(630\) 0 0
\(631\) 20.0000i 0.796187i 0.917345 + 0.398094i \(0.130328\pi\)
−0.917345 + 0.398094i \(0.869672\pi\)
\(632\) 0 0
\(633\) −25.7196 + 25.7196i −1.02226 + 1.02226i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −18.7083 + 18.7083i −0.741249 + 0.741249i
\(638\) 0 0
\(639\) 24.0000 0.949425
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −3.74166 + 3.74166i −0.147557 + 0.147557i −0.777026 0.629469i \(-0.783272\pi\)
0.629469 + 0.777026i \(0.283272\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −14.6969 + 14.6969i −0.577796 + 0.577796i −0.934296 0.356499i \(-0.883970\pi\)
0.356499 + 0.934296i \(0.383970\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 36.6606 1.43684
\(652\) 0 0
\(653\) −33.6749 33.6749i −1.31780 1.31780i −0.915513 0.402288i \(-0.868215\pi\)
−0.402288 0.915513i \(-0.631785\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 24.4949 + 24.4949i 0.955637 + 0.955637i
\(658\) 0 0
\(659\) 13.7477i 0.535535i −0.963484 0.267768i \(-0.913714\pi\)
0.963484 0.267768i \(-0.0862860\pi\)
\(660\) 0 0
\(661\) 9.16515i 0.356483i −0.983987 0.178242i \(-0.942959\pi\)
0.983987 0.178242i \(-0.0570408\pi\)
\(662\) 0 0
\(663\) −17.1464 17.1464i −0.665912 0.665912i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 44.8999 + 44.8999i 1.73853 + 1.73853i
\(668\) 0 0
\(669\) −18.3303 −0.708690
\(670\) 0 0
\(671\) 42.0000i 1.62139i
\(672\) 0 0
\(673\) −24.4949 + 24.4949i −0.944209 + 0.944209i −0.998524 0.0543149i \(-0.982703\pi\)
0.0543149 + 0.998524i \(0.482703\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22.4499 22.4499i 0.862821 0.862821i −0.128843 0.991665i \(-0.541126\pi\)
0.991665 + 0.128843i \(0.0411265\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 42.0000 1.60944
\(682\) 0 0
\(683\) −5.61249 + 5.61249i −0.214756 + 0.214756i −0.806284 0.591528i \(-0.798525\pi\)
0.591528 + 0.806284i \(0.298525\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 34.2929 34.2929i 1.30835 1.30835i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 13.7477 0.522988 0.261494 0.965205i \(-0.415785\pi\)
0.261494 + 0.965205i \(0.415785\pi\)
\(692\) 0 0
\(693\) −44.8999 44.8999i −1.70561 1.70561i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 11.0227 + 11.0227i 0.417515 + 0.417515i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 45.8258i 1.73081i 0.501069 + 0.865407i \(0.332940\pi\)
−0.501069 + 0.865407i \(0.667060\pi\)
\(702\) 0 0
\(703\) −17.1464 17.1464i −0.646690 0.646690i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −22.4499 22.4499i −0.844317 0.844317i
\(708\) 0 0
\(709\) 36.6606 1.37682 0.688409 0.725323i \(-0.258309\pi\)
0.688409 + 0.725323i \(0.258309\pi\)
\(710\) 0 0
\(711\) 40.0000i 1.50012i
\(712\) 0 0
\(713\) −19.5959 + 19.5959i −0.733873 + 0.733873i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 11.2250 11.2250i 0.419204 0.419204i
\(718\) 0 0
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) −13.0958 + 13.0958i −0.487038 + 0.487038i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −12.2474 + 12.2474i −0.454233 + 0.454233i −0.896757 0.442524i \(-0.854083\pi\)
0.442524 + 0.896757i \(0.354083\pi\)
\(728\) 0 0
\(729\) 41.0000i 1.51852i
\(730\) 0 0
\(731\) 9.16515 0.338985
\(732\) 0 0
\(733\) −18.7083 18.7083i −0.691006 0.691006i 0.271447 0.962453i \(-0.412498\pi\)
−0.962453 + 0.271447i \(0.912498\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −8.57321 8.57321i −0.315798 0.315798i
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 64.1561i 2.35683i
\(742\) 0 0
\(743\) −9.79796 9.79796i −0.359452 0.359452i 0.504159 0.863611i \(-0.331803\pi\)
−0.863611 + 0.504159i \(0.831803\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −22.4499 22.4499i −0.821401 0.821401i
\(748\) 0 0
\(749\) −27.4955 −1.00466
\(750\) 0 0
\(751\) 22.0000i 0.802791i −0.915905 0.401396i \(-0.868525\pi\)
0.915905 0.401396i \(-0.131475\pi\)
\(752\) 0 0
\(753\) 8.57321 8.57321i 0.312425 0.312425i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −7.48331 + 7.48331i −0.271986 + 0.271986i −0.829899 0.557913i \(-0.811602\pi\)
0.557913 + 0.829899i \(0.311602\pi\)
\(758\) 0 0
\(759\) 84.0000 3.04901
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 0 0
\(763\) −22.4499 + 22.4499i −0.812743 + 0.812743i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 35.0000i 1.26213i −0.775729 0.631066i \(-0.782618\pi\)
0.775729 0.631066i \(-0.217382\pi\)
\(770\) 0 0
\(771\) −18.3303 −0.660150
\(772\) 0 0
\(773\) 11.2250 + 11.2250i 0.403734 + 0.403734i 0.879547 0.475813i \(-0.157846\pi\)
−0.475813 + 0.879547i \(0.657846\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −34.2929 34.2929i −1.23025 1.23025i
\(778\) 0 0
\(779\) 41.2432i 1.47769i
\(780\) 0 0
\(781\) 27.4955i 0.983865i
\(782\) 0 0
\(783\) −17.1464 17.1464i −0.612763 0.612763i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 18.7083 + 18.7083i 0.666878 + 0.666878i 0.956992 0.290114i \(-0.0936931\pi\)
−0.290114 + 0.956992i \(0.593693\pi\)
\(788\) 0 0
\(789\) −54.9909 −1.95773
\(790\) 0 0
\(791\) 6.00000i 0.213335i
\(792\) 0 0
\(793\) −34.2929 + 34.2929i −1.21778 + 1.21778i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −22.4499 + 22.4499i −0.795218 + 0.795218i −0.982337 0.187119i \(-0.940085\pi\)
0.187119 + 0.982337i \(0.440085\pi\)
\(798\) 0 0
\(799\) 6.00000 0.212265
\(800\) 0 0
\(801\) −60.0000 −2.12000
\(802\) 0 0
\(803\) −28.0624 + 28.0624i −0.990302 + 0.990302i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 34.2929 34.2929i 1.20717 1.20717i
\(808\) 0 0
\(809\) 18.0000i 0.632846i −0.948618 0.316423i \(-0.897518\pi\)
0.948618 0.316423i \(-0.102482\pi\)
\(810\) 0 0
\(811\) 36.6606 1.28733 0.643664 0.765308i \(-0.277413\pi\)
0.643664 + 0.765308i \(0.277413\pi\)
\(812\) 0 0
\(813\) −14.9666 14.9666i −0.524903 0.524903i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −17.1464 17.1464i −0.599878 0.599878i
\(818\) 0 0
\(819\) 73.3212i 2.56205i
\(820\) 0 0
\(821\) 54.9909i 1.91920i 0.281375 + 0.959598i \(0.409210\pi\)
−0.281375 + 0.959598i \(0.590790\pi\)
\(822\) 0 0
\(823\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −5.61249 5.61249i −0.195165 0.195165i 0.602758 0.797924i \(-0.294068\pi\)
−0.797924 + 0.602758i \(0.794068\pi\)
\(828\) 0 0
\(829\) −45.8258 −1.59159 −0.795797 0.605563i \(-0.792948\pi\)
−0.795797 + 0.605563i \(0.792948\pi\)
\(830\) 0 0
\(831\) 70.0000i 2.42827i
\(832\) 0 0
\(833\) −6.12372 + 6.12372i −0.212174 + 0.212174i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7.48331 7.48331i 0.258661 0.258661i
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 55.0000 1.89655
\(842\) 0 0
\(843\) −11.2250 + 11.2250i −0.386609 + 0.386609i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 24.4949 24.4949i 0.841655 0.841655i
\(848\) 0 0
\(849\) 35.0000i 1.20120i
\(850\) 0 0
\(851\) 36.6606 1.25671
\(852\) 0 0
\(853\) −7.48331 7.48331i −0.256224 0.256224i 0.567293 0.823516i \(-0.307991\pi\)
−0.823516 + 0.567293i \(0.807991\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −11.0227 11.0227i −0.376528 0.376528i 0.493320 0.869848i \(-0.335783\pi\)
−0.869848 + 0.493320i \(0.835783\pi\)
\(858\) 0 0
\(859\) 4.58258i 0.156355i 0.996939 + 0.0781777i \(0.0249102\pi\)
−0.996939 + 0.0781777i \(0.975090\pi\)
\(860\) 0 0
\(861\) 82.4864i 2.81113i
\(862\) 0 0
\(863\) −12.2474 12.2474i −0.416908 0.416908i 0.467229 0.884137i \(-0.345253\pi\)
−0.884137 + 0.467229i \(0.845253\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 26.1916 + 26.1916i 0.889513 + 0.889513i
\(868\) 0 0
\(869\) 45.8258 1.55453
\(870\) 0 0
\(871\) 14.0000i 0.474372i
\(872\) 0 0
\(873\) −19.5959 + 19.5959i −0.663221 + 0.663221i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −14.9666 + 14.9666i −0.505387 + 0.505387i −0.913107 0.407720i \(-0.866324\pi\)
0.407720 + 0.913107i \(0.366324\pi\)
\(878\) 0 0
\(879\) 42.0000 1.41662
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 20.5791 20.5791i 0.692542 0.692542i −0.270248 0.962791i \(-0.587106\pi\)
0.962791 + 0.270248i \(0.0871058\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −39.1918 + 39.1918i −1.31593 + 1.31593i −0.398968 + 0.916965i \(0.630632\pi\)
−0.916965 + 0.398968i \(0.869368\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 22.9129 0.767610
\(892\) 0 0
\(893\) −11.2250 11.2250i −0.375629 0.375629i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −68.5857 68.5857i −2.29001 2.29001i
\(898\) 0 0
\(899\) 36.6606i 1.22270i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −34.2929 34.2929i −1.14119 1.14119i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 3.74166 + 3.74166i 0.124240 + 0.124240i 0.766493 0.642253i \(-0.222000\pi\)
−0.642253 + 0.766493i \(0.722000\pi\)
\(908\) 0 0
\(909\) −36.6606 −1.21596
\(910\) 0 0
\(911\) 36.0000i 1.19273i 0.802712 + 0.596367i \(0.203390\pi\)
−0.802712 + 0.596367i \(0.796610\pi\)
\(912\) 0 0
\(913\) 25.7196 25.7196i 0.851196 0.851196i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −44.8999 + 44.8999i −1.48272 + 1.48272i
\(918\) 0 0
\(919\) 52.0000 1.71532 0.857661 0.514216i \(-0.171917\pi\)
0.857661 + 0.514216i \(0.171917\pi\)
\(920\) 0 0
\(921\) 35.0000 1.15329
\(922\) 0 0
\(923\) 22.4499 22.4499i 0.738949 0.738949i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 9.79796 9.79796i 0.321807 0.321807i
\(928\) 0 0
\(929\) 42.0000i 1.37798i −0.724773 0.688988i \(-0.758055\pi\)
0.724773 0.688988i \(-0.241945\pi\)
\(930\) 0 0
\(931\) 22.9129 0.750939
\(932\) 0 0
\(933\) 33.6749 + 33.6749i 1.10247 + 1.10247i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.22474 1.22474i −0.0400107 0.0400107i 0.686818 0.726829i \(-0.259007\pi\)
−0.726829 + 0.686818i \(0.759007\pi\)
\(938\) 0 0
\(939\) 54.9909i 1.79456i
\(940\) 0 0
\(941\) 9.16515i 0.298775i −0.988779 0.149388i \(-0.952270\pi\)
0.988779 0.149388i \(-0.0477302\pi\)
\(942\) 0 0
\(943\) 44.0908 + 44.0908i 1.43579 + 1.43579i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 33.6749 + 33.6749i 1.09429 + 1.09429i 0.995065 + 0.0992225i \(0.0316355\pi\)
0.0992225 + 0.995065i \(0.468364\pi\)
\(948\) 0 0
\(949\) 45.8258 1.48757
\(950\) 0 0
\(951\) 84.0000i 2.72389i
\(952\) 0 0
\(953\) −8.57321 + 8.57321i −0.277714 + 0.277714i −0.832196 0.554482i \(-0.812916\pi\)
0.554482 + 0.832196i \(0.312916\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 78.5748 78.5748i 2.53996 2.53996i
\(958\) 0 0
\(959\) 54.0000 1.74375
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) −22.4499 + 22.4499i −0.723439 + 0.723439i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 24.4949 24.4949i 0.787703 0.787703i −0.193414 0.981117i \(-0.561956\pi\)
0.981117 + 0.193414i \(0.0619562\pi\)
\(968\) 0 0
\(969\) 21.0000i 0.674617i
\(970\) 0 0
\(971\) −22.9129 −0.735309 −0.367655 0.929962i \(-0.619839\pi\)
−0.367655 + 0.929962i \(0.619839\pi\)
\(972\) 0 0
\(973\) 11.2250 + 11.2250i 0.359856 + 0.359856i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 28.1691 + 28.1691i 0.901210 + 0.901210i 0.995541 0.0943306i \(-0.0300711\pi\)
−0.0943306 + 0.995541i \(0.530071\pi\)
\(978\) 0 0
\(979\) 68.7386i 2.19690i
\(980\) 0 0
\(981\) 36.6606i 1.17048i
\(982\) 0 0
\(983\) −31.8434 31.8434i −1.01565 1.01565i −0.999876 0.0157700i \(-0.994980\pi\)
−0.0157700 0.999876i \(-0.505020\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −22.4499 22.4499i −0.714590 0.714590i
\(988\) 0 0
\(989\) 36.6606 1.16574
\(990\) 0 0
\(991\) 2.00000i 0.0635321i −0.999495 0.0317660i \(-0.989887\pi\)
0.999495 0.0317660i \(-0.0101131\pi\)
\(992\) 0 0
\(993\) −60.0125 + 60.0125i −1.90444 + 1.90444i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −7.48331 + 7.48331i −0.236999 + 0.236999i −0.815606 0.578607i \(-0.803596\pi\)
0.578607 + 0.815606i \(0.303596\pi\)
\(998\) 0 0
\(999\) −14.0000 −0.442940
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1600.2.o.k.607.2 yes 8
4.3 odd 2 1600.2.o.f.607.3 yes 8
5.2 odd 4 1600.2.o.f.543.4 yes 8
5.3 odd 4 1600.2.o.f.543.1 8
5.4 even 2 inner 1600.2.o.k.607.3 yes 8
8.3 odd 2 1600.2.o.f.607.1 yes 8
8.5 even 2 inner 1600.2.o.k.607.4 yes 8
20.3 even 4 inner 1600.2.o.k.543.4 yes 8
20.7 even 4 inner 1600.2.o.k.543.1 yes 8
20.19 odd 2 1600.2.o.f.607.2 yes 8
40.3 even 4 inner 1600.2.o.k.543.2 yes 8
40.13 odd 4 1600.2.o.f.543.3 yes 8
40.19 odd 2 1600.2.o.f.607.4 yes 8
40.27 even 4 inner 1600.2.o.k.543.3 yes 8
40.29 even 2 inner 1600.2.o.k.607.1 yes 8
40.37 odd 4 1600.2.o.f.543.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1600.2.o.f.543.1 8 5.3 odd 4
1600.2.o.f.543.2 yes 8 40.37 odd 4
1600.2.o.f.543.3 yes 8 40.13 odd 4
1600.2.o.f.543.4 yes 8 5.2 odd 4
1600.2.o.f.607.1 yes 8 8.3 odd 2
1600.2.o.f.607.2 yes 8 20.19 odd 2
1600.2.o.f.607.3 yes 8 4.3 odd 2
1600.2.o.f.607.4 yes 8 40.19 odd 2
1600.2.o.k.543.1 yes 8 20.7 even 4 inner
1600.2.o.k.543.2 yes 8 40.3 even 4 inner
1600.2.o.k.543.3 yes 8 40.27 even 4 inner
1600.2.o.k.543.4 yes 8 20.3 even 4 inner
1600.2.o.k.607.1 yes 8 40.29 even 2 inner
1600.2.o.k.607.2 yes 8 1.1 even 1 trivial
1600.2.o.k.607.3 yes 8 5.4 even 2 inner
1600.2.o.k.607.4 yes 8 8.5 even 2 inner